Natural Deduction in forall x systems

This document gives a short description of how Carnap presents the systems of natural deduction from P.D. Magnus' forall x and from the Calgary remix of forall x. At least some prior familiarity with Fitch-style proof systems is assumed.

Propositional Systems

Notation

The different admissible keyboard abbreviations for the different connectives are as follows:

Connective

Keyboard

→

->, =>,>

∧

/\, &, and

∨

\/, |, or

↔

<->, <=>

¬

-, ~, not

⊥

!?

The available sentence letters are A through Z, together with the infinitely many subscripted letters P1, P2, … written P_1, P_2 and so on.

Proofs consist of a series of lines. A line is either an assertion line containing a formula followed by a : and then a justification for that formula, or a separator line containing two dashes, thus: --. A justification consists of a rule abbreviation followed by zero or more numbers (citations of particular lines) and pairs of numbers separated by a dash (citations of a subproof contained within the given line range).

A subproof is begun by increasing the indentation level. The first line of a subproof should be more indented than the containing proof, and the lines directly contained in this subproof should maintain this indentation level. (Lines indirectly contained, by being part of a sub-sub-proof, will need to be indented more deeply.) The subproof ends when the indentation level of the containing proof is resumed; hence, two contiguous sub-proofs of the same containing proof can be distinguished from one another by inserting a separator line between them at the same level of indentation as the containing proof. The final unindented line of a derivation will serve as the conclusion of the entire derivation.

Here's an example derivation, using system SL of P.D. Magnus forall x:

Playground

P:AS
P:AS
P:R 2
P->P:->I 2-3
P->(P->P):->I 1-4

Basic Rules

forall x System SL

The minimal system SL for P.D. Magnus' forall x (the system used in a proofchecker constructed with .ForallxSL in Carnap's Pandoc Markup) has the following set of rules for direct inferences:

Rule

Abbreviation

Premises

Conclusion

And-Elim.

∧E

φ ∧ ψ

φ/ψ

And-Intro.

∧I

φ, ψ

φ ∧ ψ

Or-Elim

∨E

¬ψ, φ ∨ ψ

φ

¬φ, φ ∨ ψ

ψ

Or-Intro

∨I

φ/ψ

φ ∨ ψ

Condtional-Elim

→E

φ, φ → ψ

ψ

Biconditional-Elim

↔E

φ/ψ, φ ↔ ψ

ψ/φ

Reiteration

R

φ

φ

We also have four rules for indirect inferences:

→I, which justifies an assertion of the form φ → ψ by citing a subproof beginning with the assumption φ and ending with the conclusion ψ;

↔I, which justifies an assertion of the form φ↔ψ by citing two subproofs, beginning with assuptions φ, ψ, respectively, and ending with conclusions ψ, φ, respectively;

¬I, which justifies an assertion of the form ¬φ by citing a subproof beginning with the assumption φ and ending with a pair of lines ψ,¬ψ.

¬E, which justifies an assertion of the form φ by citing a subproof beginning with the assumption ¬φ and ending with a pair of lines ψ,¬ψ.

Finally, PR can be used to justify a line asserting a premise, and AS can be used to justify a line making an assumption. Assumptions are only allowed on the first line of a subproof.

forall x System SL Plus

The extended system SL Plus for P.D. Magnus' forall x (the system used in a proofchecker constructed with .ForallxSLPlus in Carnap's Pandoc Markup) also adds the following rules:

Rule

Abbreviation

Premises

Conclusion

Dilemma

DIL

φ ∨ ψ, φ → χ, ψ → χ

χ

Hypothetical Syllogism

HS

φ → ψ, ψ → χ

φ → χ

Modus Tollens

MT

φ → ψ, ¬ψ

¬φ

As well as the following exchange rules, which can be used within a propositional context Φ:

Rule

Abbreviation

Premises

Conclusion

Commutativity

Comm

Φ(φ ∧ ψ)

Φ(ψ ∧ φ)

Φ(φ ∨ ψ)

Φ(ψ ∨ φ)

Φ(φ ↔ ψ)

Φ(ψ ↔ φ)

Double Negation

DN

Φ(φ)/Φ(¬¬φ)

Φ(¬¬φ)/Φ(φ)

Material Conditional

MC

Φ(φ → ψ)

Φ(¬φ ∨ ψ)

Φ(¬φ ∨ ψ)

Φ(φ → ψ)

Φ(φ ∨ ψ)

Φ(¬φ → ψ)

Φ(¬φ → ψ)

Φ(φ ∨ ψ)

BiConditional Exchange

↔ex

Φ(φ ↔ ψ)

Φ(φ → ψ ∧ ψ → φ)

Φ(φ → ψ ∧ ψ → φ)

Φ(φ ↔ ψ)

DeMorgan's Laws

DeM

Φ(¬(φ ∧ ψ))

Φ(¬φ ∨ ¬ψ)

Φ(¬(φ ∨ ψ))

Φ(¬φ ∧ ¬ψ)

Φ(¬φ ∨ ¬ψ)

Φ(¬(φ ∧ ψ))

Φ(¬φ ∧ ¬ψ)

Φ(¬(φ ∨ ψ))

Calgary TFL

The system TFL from the Calgary Remix of forall x (the system used in a proofchecker constructed with .ZachTFL in Carnap's Pandoc Markup) has the following set of rules for direct inferences:

Rule

Abbreviation

Premises

Conclusion

And-Elim.

∧E

φ ∧ ψ

φ/ψ

And-Intro.

∧I

φ, ψ

φ ∧ ψ

Or-Intro

∨I

φ/ψ

φ ∨ ψ

Negation-Elim

¬E

φ, ¬φ

⊥

Explosion

X

⊥

ψ

Biconditional-Elim

↔E

φ/ψ, φ ↔ ψ

ψ/φ

Reiteration

R

φ

φ

Disjunctive Syllogism

DS

¬ψ/¬φ, φ ∨ ψ

φ/ψ

Modus Tollens

MT

φ → ψ, ¬ψ

¬φ

Double Negation Elim.

DNE

¬¬φ

φ

DeMorgan's Laws

DeM

¬(φ ∧ ψ)

¬φ ∨ ¬ψ

¬(φ ∨ ψ)

¬φ ∧ ¬ψ

¬φ ∨ ¬ψ

¬(φ ∧ ψ)

¬φ ∧ ¬ψ

¬(φ ∨ ψ)

We also have five rules for indirect inferences:

→I, which justifies an assertion of the form φ → ψ by citing a subproof beginning with the assumption φ and ending with the conclusion ψ;

↔I, which justifies an assertion of the form φ ↔ ψ by citing two subproofs, beginning with assumptions φ, ψ, respectively, and ending with conclusions ψ, φ, respectively;

¬I, which justifies an assertion of the form ¬φ by citing a subproof beginning with the assumption φ and ending with a conclusion ⊥.

∨E, which justifies an assertion of the form φ by citing a disjunction ψ ∨ χ and two subproofs beginning with assumptions ψ, χ respectively and each ending with the conclusion φ.

IP (indirect proof), which justifies an assertion of the form φ by citing a subproof beginning with the assumption ¬φ and ending with a conclusion ⊥.

LEM (Law of the Excluded Middle), which justifies an assertion of the form ψ by citing two subproofs beginning with assumptions φ, ¬φ respectively and each ending with the conclusion ψ.

As above, PR can be used to justify a line asserting a premise, and AS can be used to justify a line making an assumption. Assumptions are only allowed on the first line of a subproof.

First-Order Systems

Notation

The different admissible keyboard abbreviations for quantifiers and equality is as follows:

Connective

Keyboard

∀

A

∃

E

=

=

The forall x first-order systems do not contain sentence letters.

Application of a relation symbol is indicated by directly appending the arguments to the symbol.

The available relation symbols are A through Z, together with the infinitely many subscripted letters F1, F2, … written F_1, F_2,…. The arity of a relation symbol is determined from context.

The available constants are a through w, with the infinitely many subscripted letters c1, c2, … written c_1, c_2,….

The available variables are x through z, with the infinitely many subscripted letters x1, x2, … written x_1, x_2,….

Basic Rules

The first-order forall x systems QL and FOL (the systems used in proofcheckers constructed with .ForallxQL, and ZachFOL respectively) extend the rules of the system SL and TFL respectively with the following set of new basic rules:

Rule

Abbreviation

Premises

Conclusion

Existential Introduction

∃E

φ(σ)

∃xφ(x)

Universal Elimination

∀E

∀xφ(x)

φ(σ)

Universal Introduction

∀E

φ(σ)

∀xφ(x)

Equality Introduction

=I

σ = σ

Equality Elimination

=E

σ = τ, φ(σ)/φ(τ)

φ(τ)/φ(σ)

Where Universal Introduction is subject to the restriction that σ must not appear in φ(x), or any undischarged assumption or in any premise of the proof.1

It also adds one new rule for indirect derivations: ∃E, which justifies an assertion ψ by citing an assertion of the form ∃xφ(x) and a subproof beginning with the assumption φ(σ) and ending with the conclusion ψ, where σ does not appear in ψ, ∃xφ(x), or in any of the undischarged assumptions or premises of the proof.

The Calgary remix system FOL adds, in addition to these rules, the rule

Rule

Abbreviation

Premises

Conclusion

Conversion of Quantifiers

CQ

¬∀xφ(x)

∃x¬φ(x)

∃x¬φ(x)

¬∀xφ(x)

¬∃xφ(x)

∀x¬φ(x)

∀x¬φ(x)

¬∃xφ(x)

Technically, Carnap checks only the assumptions and premises that are used in the derivation of φ(σ). This has the same effect in terms of what's derivable, but is a little more lenient.↩